Introduction to Financial Mathematics

  • Summer schools
  • Department of Mathematics
  • Application code SS-ME302
  • Starting 2020
  • Short course: Open
  • Location: Houghton Street, London

Determining rational prices of financial contracts, so-called financial derivatives, is a key question in financial mathematics. This course introduces a range of mathematical concepts and techniques for the modelling of financial markets in both discrete and continuous time that allow us to investigate this problem. The pricing and hedging of financial derivatives in the binomial tree and the Black-Scholes models are studied in detail.

The course starts with the necessary probability background for modelling uncertainty. This includes a study of random variables, distribution functions, expectation, independence and conditional expectation. Sigma algebras are introduced to model information in the context of dynamic trading. All mathematical concepts and techniques are illustrated by examples.

Then, the course introduces the concepts of self-financing trading, no-arbitrage, and pricing by replication in the context of the binomial tree model. The no-arbitrage prices are expressed as expectations with respect to the martingale probability measure. This enables the computation of no-arbitrage prices and hedging strategies. Examples of pricing and hedging of several financial derivatives are worked out in detail.

Next, the course moves to the study of continuous time models. The standard Brownian motion as well as some elements of stochastic analysis and martingale theory are introduced. Stochastic integration with respect to a Brownian motion and Itô calculus are developed to model dynamic trading in continuous time. Filtrations are introduced as models for the propagation of information in financial markets. Furthermore, Girsanov's theorem is studied as a tool for changing from the historical probability measure to the pricing measure, the so-called risk-neutral probability measure.

Building on this background, the Black & Scholes option pricing theory is developed. The no-arbitrage prices of financial derivatives are represented as expectations with respect to the risk-neutral probability measure of the derivatives’ discounted payoffs. This leads to practical pricing formulas such as the Black-Scholes formula and allows the computations of no arbitrage. It is also shown how no-arbitrage prices and hedging strategies can be computed by solving the Black & Scholes partial differential equation with suitable boundary conditions. The sensitivities of no-arbitrage prices with respect to model parameters, namely, the so-called Greeks, such as the delta, the gamma, the vega, etc, are also studied.

Session: Three
Dates: 3 August – 21 August 2020
Lecturers: Dr Christoph Czichowsky and Professor Johannes Ruf


Programme details

Key facts

Level: 300 level. Read more information on levels in our FAQs

Fees:  Please see Fees and payments

Lectures: 36 hours 

Classes: 18 hours

Assessment*: Two written examinations

Typical credit**: 3-4 credits (US) 7.5 ECTS points (EU)

*Assessment is optional

**You will need to check with your home institution

For more information on exams and credit, read Teaching and assessment


Calculus and Probability at intermediate undergraduate level, Real Analysis at introductory undergraduate level.

Programme structure

  • Probability:  probability spaces, random variables, expectation, conditional expectation and stochastic processes
  • Binomial tree model: no arbitrage, self-financing portfolios, replication, martingale probability measures, pricing and hedging of derivatives
  • Stochastic calculus: martingales, Brownian motion, stochastic integration, Ito’s formula, changes of probability measures and Girsanov’s theorem
  • Black & Scholes option pricing theory: characterisation and replication of attainable payoffs, Black-Scholes formula, Black-Scholes partial differential equation

Course outcomes

Students obtain mathematical background for the modelling of financial markets. This enables them to analyse mathematical questions arising in the context of financial markets, such as the pricing and hedging of derivatives.


The LSE Department of Mathematics is internationally recognised for its teaching and research. Located within a world-class social science institution, the department aims to be a leading centre for Mathematics in the Social Sciences. The Department of Mathematics was submitted jointly to REF 2014 with LSE's Department of Statistics: 84% of the research outputs of the two departments were classed as either world-leading or internationally excellent in terms of originality, significance and rigour.

The Department has more than doubled in size over the past few years, and this growth trajectory reflects the increasing impact that mathematical theory and mathematical techniques are having on subjects such as economics and finance, and on many other areas of the Social Sciences.

On this three week intensive programme, you will engage with and learn from full-time lecturers from the LSE’s mathematics faculty.

Reading materials

-   S. E. Shreve, Stochastic Calculus for Finance.  Volume I: The Binomial Asset Pricing Model; Volume II: Continuous-Time Models, Springer, 2004. [required]
-   J. Hull, Options, Futures and Other Derivatives, 9th edition, Pearson, 2015. [recommended]
-   S. Ross, Introduction to Probability Models, 10th edition, Academic Press, 2010. [recommended]
-   P. Wilmott,  S. Howison  and  J. Dewynne,  The  Mathematics  of  Financial  Derivatives, CUP, 1995. [recommended]

*A more detailed reading list will be supplied prior to the start of the programme

**Course content, faculty and dates may be subject to change without prior notice

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